What is the name for the following categorical property? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T04:14:26Z http://mathoverflow.net/feeds/question/8244 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8244/what-is-the-name-for-the-following-categorical-property What is the name for the following categorical property? J. H. S. 2009-12-08T20:35:39Z 2009-12-21T18:19:22Z <p>Is there a name for those categories where objects posses a given structure and every bijective morphism determines an isomorphism between the corresponding objects?</p> <p>Examples of categories of that type abound: <strong>Gr</strong>, <strong>Set</strong>, ...</p> <p>An specific example of a category where the constraint doesn't hold is given by <strong>Top</strong>: a morphism there is a continuous function between topological spaces. Now, it is easy to give <em>here</em> a concrete example of a bijective morphism between [0,1) and $\mathbb{S}^{1}$ that fails to be an isomorphism of topological spaces. In fact, much more is known in this case, right?</p> http://mathoverflow.net/questions/8244/what-is-the-name-for-the-following-categorical-property/8250#8250 Answer by Tom Leinster for What is the name for the following categorical property? Tom Leinster 2009-12-08T21:11:28Z 2009-12-08T21:16:29Z <p>The comments on the question point out that it's not really well-posed: the property "bijective" isn't defined for morphisms of an arbitrary category. </p> <p>However, for maps between sets, "bijective" means "injective and surjective". A common way to interpret "injective" in an arbitrary category is "monic", and a common way to interpret "surjective" in an arbitrary category is "epic". So we might interpret "bijective" as "monic and epic".</p> <p>Then JHS's question becomes: is there a name for categories in which every morphism that is both monic and epic is an isomorphism? And the answer is yes: <a href="http://ncatlab.org/nlab/show/balanced+category" rel="nofollow">balanced</a>.</p> <p>It's not a particularly inspired choice of name, nor does it seem to be a particularly important concept. But the terminology is quite old and well-established, in its own small way.</p> <p>Incidentally, you don't <i>have</i> to interpret "injective" and "surjective" in the ways suggested. You could, for instance, interpret "surjective" as "regular epic", and indeed that's often a sensible thing to do. But then the question becomes trivial, since any morphism that's both monic and regular epic is automatically an isomorphism.</p> http://mathoverflow.net/questions/8244/what-is-the-name-for-the-following-categorical-property/8310#8310 Answer by Evgeny Shinder for What is the name for the following categorical property? Evgeny Shinder 2009-12-09T04:10:00Z 2009-12-09T16:24:18Z <p>As other people commented, the language of categories is richer than the language of sets with structures (Bourbaki structures). There are many categories, where objects don't have an underlying set.</p> <p>However, one can restate the property you formulate as follows: the category <strong>C</strong> admits a <a href="http://ncatlab.org/nlab/show/faithful+functor" rel="nofollow">faithful</a> <a href="http://ncatlab.org/nlab/show/conservative+functor" rel="nofollow">conservative</a> functor to <strong>Sets</strong>. Then we can interpret the fiber of this functor over a given set S as the set of structures on S and call the functor a forgetful functor. By faithfulness the homs in <strong>C</strong> will be subsets of those in <strong>Sets</strong>, and we can say that the homs in <strong>C</strong> preserve the structure.</p> http://mathoverflow.net/questions/8244/what-is-the-name-for-the-following-categorical-property/8402#8402 Answer by Emily Riehl for What is the name for the following categorical property? Emily Riehl 2009-12-10T00:30:20Z 2009-12-10T00:30:20Z <p>This isn't quite the question you asked, but does address the notion of ''bijective'' morphisms in categories, so I hope you'll forgive this digression.</p> <p>The examples you've mentioned - <strong>Set</strong>, <strong>Gp</strong>, <strong>Top</strong> - are all <em>concrete</em>, meaning they are equipped with a forgetful functor <em>U</em> to <strong>Set</strong>. We say a morphism <em>f</em> in a concrete category <em>C</em> is <em>injective</em> if its image <em>Uf</em> is injective, i.e., monic in the category <strong>Set</strong>. Dually, <em>f</em> is <em>surjective</em> if <em>Uf</em> is surjective. One usually thinks of concrete categories as "sets with structure", so these definitions coincide with the common use of such terminology: e.g., we call a map of spaces surjective when the underlying map of sets is.</p> <p>So we have four adjectives to use for arrows in <em>C</em>: monic, epic, injective, surjective. It's an easy exercise to see that all injections are monic and all surjections are epic. The converse is not true in general, but finding examples of monos that aren't injective and epis that aren't surjective can be tricky, and here's why.</p> <p>Often, particularly in ''algebraic'' examples, the functor <em>U</em> : <em>C</em> &rarr; <strong>Set</strong> has a left adjoint <em>F</em>. When this is the case, it is an easy exercise to see that every mono must be injective. Dually, if <em>U</em> has a right adjoint, then every epi is surjective. So for example, the forgetful functor <em>U</em> : <strong>Top</strong> &rarr; <strong>Set</strong> has both adjoints, and hence for spaces the notions injective/surjective and monic/epic coincide, at which point Tom's post answers your question.</p> <p>Here are some examples of concrete categories where these concepts differ, all of which can be found in Francis Borceux's <em>Handbook of Categorical Algebra</em> (I think). In the category of divisible abelian groups, the quotient map $\mathbb{Q} \rightarrow \mathbb{Q}/\mathbb{Z}$ is monic, though it's clearly not injective. In the category of monoids, the inclusion $\mathbb{N} \rightarrow \mathbb{Z}$ is epic, though not surjective. In the category of Hausdorff spaces, the epis are continuous functions with dense image, so also need not be surjective.</p> http://mathoverflow.net/questions/8244/what-is-the-name-for-the-following-categorical-property/9496#9496 Answer by Paul Taylor for What is the name for the following categorical property? Paul Taylor 2009-12-21T18:19:22Z 2009-12-21T18:19:22Z <p>"every bijective morphism determines an isomorphism"</p> <p>I think you mean that <i>the forgetful functor reflects invertibility</i>.</p> <p>Let $\bf A$ be the "category of objects with structure" and their structure-preserving maps (homomorphisms), $\bf S$ the category of carriers (maybe sets and functions) and $U:{\bf A}\to{\bf S}$ the "forgetful" functor between them. In fact, just let $U:{\bf A}\to{\bf S}$ be any functor you like.</p> <p>Now let $f:X\to Y$ be any morphism of $\bf A$. You are saying that, whenever $U f:U X\to U Y$ is an isomorphism (such as a bijection) then $f$ was already an isomorphism.</p> <p>The forgetful functor from any category of algebras has this property, as more generally does the right adjoint of any monadic adjunction. However, the underlying set functor from the usual category of topological spaces does not, because there are many different topologies that can be put on a set.</p>